Optimal. Leaf size=142 \[ \frac{4 x+3}{236196 \left (4 x^2+6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{944784}-\frac{5 \log \left (4 x^2+6 x+9\right )}{2834352}+\frac{1}{472392 (3-2 x)}-\frac{1}{157464 (2 x+3)}-\frac{\log (3-2 x)}{354294}+\frac{\log (2 x+3)}{118098}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{472392 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.267688, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{4 x+3}{236196 \left (4 x^2+6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{944784}-\frac{5 \log \left (4 x^2+6 x+9\right )}{2834352}+\frac{1}{472392 (3-2 x)}-\frac{1}{157464 (2 x+3)}-\frac{\log (3-2 x)}{354294}+\frac{\log (2 x+3)}{118098}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{472392 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(9 - 6*x + 4*x^2)/(729 - 64*x^6)^2,x]
[Out]
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Rubi in Sympy [A] time = 48.1266, size = 117, normalized size = 0.82 \[ \frac{8 x + 6}{472392 \left (4 x^{2} + 6 x + 9\right )} - \frac{\log{\left (- 2 x + 3 \right )}}{354294} + \frac{\log{\left (2 x + 3 \right )}}{118098} - \frac{\log{\left (4 x^{2} - 6 x + 9 \right )}}{944784} - \frac{5 \log{\left (4 x^{2} + 6 x + 9 \right )}}{2834352} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{1417176} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{157464} - \frac{1}{157464 \left (2 x + 3\right )} + \frac{1}{472392 \left (- 2 x + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((4*x**2-6*x+9)/(-64*x**6+729)**2,x)
[Out]
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Mathematica [A] time = 0.117685, size = 111, normalized size = 0.78 \[ \frac{-3 \log \left (4 x^2-6 x+9\right )-5 \log \left (4 x^2+6 x+9\right )+\frac{648 x}{-16 x^4-24 x^3+54 x+81}-8 \log (3-2 x)+24 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2834352} \]
Antiderivative was successfully verified.
[In] Integrate[(9 - 6*x + 4*x^2)/(729 - 64*x^6)^2,x]
[Out]
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Maple [A] time = 0.022, size = 111, normalized size = 0.8 \[ -{\frac{1}{314928\,x+472392}}+{\frac{\ln \left ( 2\,x+3 \right ) }{118098}}-{\frac{1}{-1417176+944784\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{354294}}-{\frac{1}{708588} \left ( -3\,x-{\frac{9}{4}} \right ) \left ({x}^{2}+{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{5\,\ln \left ( 16\,{x}^{2}+24\,x+36 \right ) }{2834352}}+{\frac{\sqrt{3}}{157464}\arctan \left ({\frac{ \left ( 32\,x+24 \right ) \sqrt{3}}{72}} \right ) }-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{944784}}+{\frac{\sqrt{3}}{1417176}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((4*x^2-6*x+9)/(-64*x^6+729)^2,x)
[Out]
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Maxima [A] time = 1.55608, size = 128, normalized size = 0.9 \[ \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{1417176} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )}} - \frac{5}{2834352} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{118098} \, \log \left (2 \, x + 3\right ) - \frac{1}{354294} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 - 6*x + 9)/(64*x^6 - 729)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213703, size = 269, normalized size = 1.89 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 3 \, \sqrt{3}{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 24 \, \sqrt{3}{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (2 \, x + 3\right ) + 8 \, \sqrt{3}{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \log \left (2 \, x - 3\right ) - 54 \,{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) - 6 \,{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 648 \, \sqrt{3} x\right )}}{8503056 \,{\left (16 \, x^{4} + 24 \, x^{3} - 54 \, x - 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 - 6*x + 9)/(64*x^6 - 729)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.832613, size = 116, normalized size = 0.82 \[ - \frac{x}{69984 x^{4} + 104976 x^{3} - 236196 x - 354294} - \frac{\log{\left (x - \frac{3}{2} \right )}}{354294} + \frac{\log{\left (x + \frac{3}{2} \right )}}{118098} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{944784} - \frac{5 \log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{2834352} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{1417176} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{157464} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x**2-6*x+9)/(-64*x**6+729)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.219882, size = 143, normalized size = 1.01 \[ \frac{1}{157464} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{1417176} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} + 6 \, x + 9\right )}{\left (2 \, x + 3\right )}{\left (2 \, x - 3\right )}} - \frac{5}{2834352} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{944784} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{118098} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{354294} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 - 6*x + 9)/(64*x^6 - 729)^2,x, algorithm="giac")
[Out]